1:41 AM
Okay y'all, I have to figure something out about Morse theory for my class, and I sort of get it intuitively... but I actually don't know how to prove it.
The statement is the following: Given a closed curve C in R^2, there is some line L in R^2 such that the function F:C-->R given by F(x)={distance from x to the line L} is a Morse function.
Is this true? I really feel like it should be.

1 hour later…
2:43 AM
Oh apparently this follows from Sard's theorem... Oh my...

6 hours later…
8:19 AM
@user40276 for perfect stacks, i'd definitely recommend reading that ben-zvi--francis--nadler paper, they'll be able to tell you much more than i can about all those sorts of issues
@user40276 as you say, it's not hard to check that our definition reduces to the cyclic bar complex in the case of a one-object spectrally-enriched $\infty$-category, i.e. an associative (i.e. $A_\infty$) ring spectrum. this has always been the correct/expected definition, and i believe it can be made rigorous e.g. using a cofibrant representative in a "good" model category of spectra (i.e. one with a symmetric monoidal smash product, say EKMM -- i'd imagine they discuss this in their book).
i think what dundas--goodwillie--mccarthy are getting at on page 125 is the following.
1. their foundational choice of using Gamma-spaces to model connective spectra means that they can't make this cyclic bar complex definition directly, i.e. without cofibrant replacement.
2. they don't see how to get the cyclotomic structure from such a cyclic bar construction, anyways.
their way of solving these problems is to use bokstedt's alternative (but equivalent) model for THH. personally i always found this definition pretty conceptually opaque, and i think a major advantage of the nikolaus--scho
@TylerLawson does this have something to do with the "handedness" of what it means for the t-structure to be compatible with the symmetric monoidal structure?

9:12 AM
@user40276 The discussion at the beginning of section III.5 of the recent Nikolaus-Scholze paper might be helpful to orient oneself among all the various definitions of THH

9:25 AM
I have a very basic question, which I assume I don't know because of my background. If $X,Y$ are reasonable spaces (e.g. finite CW complexes), when is the suspension homomorphism $[X,Y] \rightarrow [SX,SY]$ a bijection? I know the answer when $X$ is a sphere, this is the Freudenthal suspension theorem, but what can we say in general?
What is the canonical reference for this type of stuff?

Well, it's not true even when $X$ is a sphere. You need $X$ to be small dimensional. Essentially by adjunction you are asking whether [X,-] sends the map Y→ΩΣY to a bijection. If the map is n-connected this happens when X is (n-1)-dimensional (I think? I might get this off by one), so you're asking about the connectivity of Y→ΩΣY
The keyword here is Blakers-Massey theorem applied to the homotopy pushout square giving ΣY. i don't know what the "canonical" reference for it is, though

9:43 AM
The nlab page has a very complete list of references (highlights are Rezk's 2014 paper and the generalization by Anel, Biedermann, Finster and Joyal)

Sorry what I meant of course is that if $X=S^i$ is a sphere, and $Y$ is $n-1$ connected with $i<2n-1$ then this is a bijection.

Yes, as I said it follows from the connectivity of Y→ΩΣY coming from Blakers-Massey (I believe there is an essentially equivalent perspective using the James' splitting, but BM is worth knowing all on its own)

But thanks, I will delve into this. I need to learn some of this language.
The connectivity of this map is the largest $n$ such that for all $k\leq n$ the induced map $\pi_k(Y)\rightarrow \pi_k(\Omega\Sigma Y)$ is zero? Thus if for example $Y$ is $n$ connected, this map will be at least $n$ connected as well.

9:59 AM
More or less (there's weird stuff at the boundary, you just ask the map to be surjective for $k=n$). If I recall correctly Freudenthal's suspension theorem states that the map is going to be 2n connected for n connected
The literature is slightly inconsistent on what exactly means for a map to be n-connected, up to a shift of 1
The idea is that a fibration is n-connected iff all the fibers are (n-1)-connected (sigh)

okay

2 hours later…
11:35 AM
Suppose that G is a simplicial group and K \subset H are simplicial subgroups such that K -> H is a weak equivalence. It seems to me that then G/K -> G/H is also a weak equivalence, and one can see this by comparing long exact sequences of some fibration or another (though there is always a subtlety at pi_0, which I believe can be resolved). Can I cite this somewhere?

4 hours later…
3:58 PM
@TomBachmann does this follow from a model structure on sGrp? (i don't know that it does, i'm not sure what it takes to be cofibrant / a cofibration in the standard (projective) model structure)

@AaronMazel-Gee Allowing the best possible guesses about such a model structure, how would the result follow?

4:10 PM
Offhand, the morphism G/K \to G/H is levelwise surjective, and a such a morphism of s'l groups is a fibration of the underlying s'l sets (corollary of Prop'n 1.23 of math.northwestern.edu/~pgoerss/papers/ucnotes.pdf for example). Kan-Quillen is right proper, and so the levelwise kernel is the homotopy fiber. Levelwise kernel is H/K and I would hope that H/K is a discrete space if K \to H is an equiv.

4:54 PM
@AaronMazel-Gee yes, exactly, and the monoidal structure is only compatible with being bounded below in the t-structure rather than being bounded above